Contents

Idea

For GG a group, a GG-torsor (also called a principal homogeneous space) is an inhabited object/spacePP with an actionρ:G×P→P\rho : G \times P \to P by GG that is

free: only the identity element acts trivially;

and

transitive: for every two points in (a fiber of) the space, there is an element of the group taking one to the other.

In other words, in the classical case where we are working in the category of sets over the point, a torsor is a heap:

a GG-set PP with action ρ:G×P→P\rho: G \times P \to P such that every choice of point p∈Pp \in P induces an isomorphism of GG-sets

ρ(−,p):G→≃P.
\rho(-,p) : G \stackrel{\simeq}{\to} P
\,.

This says equivalently that after picking any point of PP as the identity , PP acquires a group structure isomorphic to GG. But this is a non-canonical isomorphism: every choice of point of PP yields a different isomorphism.

As a slogan we can summarize this as: A torsor is like a group that has forgotten its neutral element.

Again, this applies to torsors “over the point” in SetSet. More generally, one may consider torsors over some base space BB (in other words, working in the topos of sheaves over BB instead of SetSet). In this case the term GG-torsor is often used more or less a synonym for the term GG-principal bundle, but torsors are generally understood in contexts much wider than the term “principal bundle” is usually taken to apply. And a principal bundle is strictly speaking a torsor that is required to be locally trivial . Thus, while the terminology ‘principal bundle’ is usually used in the setting of topological spaces or smooth manifolds, the term torsor is traditionally used in the more general contex of Grothendieck topologies (faithfully flat and étale topology in particular), topoi and for generalizations in various category-theoretic setups. While in the phrase ‘GG-principal bundle’ GG is usually a (topological) group or groupoid, when we say ‘GG-torsor’, GG is usually a presheaf or sheaf of group(oid)s, or GG is a plain category (not necessarily even a groupoid).

A GG-torsor, without any base space given, can also simply be an inhabited transitive free GG-set, which is the same as a principal GG-bundle over the point. The notion may also be defined in any category with products: a torsor over a group objectGG is a well-supported objectEE together with a GG-action α:G×E→E\alpha: G \times E \to E such that the arrow

Remark

As we explain below, a torsor is in some tautological sense locally trivial, but some care must be taken in interpreting this. One sense is that there is a cover UU of 11 (so that U→1U \to 1 is epi, i.e., UU is inhabited) such that the torsor, when pulled back to UU, becomes trivial (i.e., isomorphic to GG as GG-torsor). But this is a very general notion of “cover”. A more restrictive sense frequently encountered in the literature is that “cover” means a coproduct of subterminal objects Ui↪1U_i \hookrightarrow 1 such that U=∑iUiU = \sum_i U_i is inhabited (e.g., an open cover of a space BB seen as the terminal object of the sheaf topos Sh(B)Sh(B)), and “torsor” would then refer to the local triviality condition for some such UU. This is the more usual sense when referring to principal bundles as torsors. Or, “cover” could refer to a covering sieve in a Grothendieck topology.

(The condition on the action can be translated to give transitivity etc. in the case of BB is a point (left as a standard exercise).)

Examples

In sets

An affine space of dimension nn over a fieldkk is a torsor for the additive group knk^n: this acts by translation.

A unit of measurement is (typically) an element in an ℝ×\mathbb{R}^\times-torsor, for ℝ×\mathbb{R}^\times the multiplicative group of non-zero real numbers: for uu any unit and r∈ℝr \in \mathbb{R} any non-vanishing real number, also rur u is a unit. And for u1u_1 and u2u_2 two units, one is expressed in terms of the other by a unique r≠0r \neq 0 as u1=ru2u_1 = r u_2. For instance for units of mass we have the unit of kilogram and that of gram and there is a unique number, r=1000r = 1000 with

kg=1000g.
kg = 1000 g
\,.

In topological spaces

A topological GG-principal bundleπ:P→B\pi: P \to B is an example of a torsor over BB in TopTop. This becomes a definition of principal bundle if we demand local triviality with respect to some open cover of BB (see the remarks below).

then we get a division structure DD on PP for which pp behaves as an identity (i.e., D(x,x)=pD(x, x) = p for all x∈Px \in P), so that PP acquires a group structure isomorphic to that of GG.

Local trivialization

In other categories CC besides SetSet, we cannot just “pick a point” of PP even if P→1P \to 1 is an epimorphism, so this argument cannot be carried out, and indeed trivializations may not exist. However, it is possible to construct a local trivialization of a torsor, following a general philosophy from topos theory that a statement is “locally true” in a category CC if it becomes true when reinterpreted in a slice after pulling back C→C/UC \to C/U, where UU is inhabited. (This in some sense is the basis of Kripke-Joyal semantics.)

In the present case, we may take U=PU = P. Although we cannot “pick a point” of PP (= global section of P→1P \to 1), we can pick a point of PP if we reinterpret it by pulling back to C/PC/P. In other words, π2:P×P→1×P≅P\pi_2: P \times P \to 1 \times P \cong P does have a global section regarded as an arrow in C/PC/P. In fact, there is a “generic point”: the diagonal Δ:P→P×P\Delta: P \to P \times P. Then, we may mimic the argument above, and consider the pullback diagram

living in C/PC/P. As argued above, the vertical arrow on the left is an isomorphism; in fact, it is the isomorphism ⟨ρ,π2⟩:G×P→P×P\langle \rho, \pi_2 \rangle: G \times P \to P \times P we started with!

Thus, a GG-torsor in a category with products can be tautologically interpreted in terms of GG-actions on objects PP which become trivialized upon pulling back to the slice C/PC/P.